What are the alternative methods for integrating this function? $$f(x)=\frac {1}{(\csc (x )+ \cos (x) )} $$
I found the answer by writing it as $sin (x)/(1+\sin (x)\cos (x))$ and using trigonometric twins method $ sin (x)=1/2 ((\sin (x)+\cos(x) )+(\sin (x)-\cos (x) )$
But I want to know some alternate approaches if possible!
As Dr. Sonnhard Graubner commented, the tangent half-angle substitution is one of the most classical solution ... even if, in some cases like the one you posted, it can lead to quite unpleasant formulas.
So, using $t=\tan(\frac x2)$, we obtain $$\int\frac {dx}{\csc (x )+ \cos (x) }=\int \frac{4 t}{t^4-2 t^3+2 t^2+2 t+1}\,dt$$ The denominator does not show real roots but has the good idea to factorize $$t^4-2 t^3+2 t^2+2 t+1=\Big(t^2+\left(\sqrt{3}-1\right) t-\sqrt{3}+2\Big)\Big(t^2-\left(\sqrt{3}+1\right) t+\sqrt{3}+2\Big)$$ Now, as usual, partial fraction decomposition gives for the integrand$$-\frac{t-(\sqrt{3}+2)}{\sqrt{3} \left(t^2-(\sqrt{3}+1) t+(\sqrt{3}+2)\right)}+\frac{t+(\sqrt{3}-2)}{\sqrt{3} \left(t^2+(\sqrt{3}-1) t-(\sqrt{3}-2)\right)}$$ which let us with some standard integrals and the result is as funny as what you did show from Wolfram Alpha.